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Parabolic hysteresis problems revisited: Finite element error analysis and convergent Newton-type solvers

Shu Xu, Liqun Cao

TL;DR

This work advances the numerical analysis of parabolic PDEs with hysteresis by delivering rigorous finite element error estimates for backward Euler discretizations in both semilinear and quasilinear settings. It then introduces a robust Jacobian smoothing Newton framework, including an arc-based smoothing strategy, and proves global convergence with local quadratic rates. Comprehensive numerical experiments verify theoretical rates and demonstrate that the proposed arc-based solver outperforms classical derivative-free methods and other Newton-type approaches in speed and robustness. The results provide a practical, theoretically grounded pipeline for accurately and efficiently simulating hysteresis-influenced parabolic problems in 1–3D. The approach is applicable to a broad class of hysteresis models, including linear play and Preisach operators, with potential impact on engineering simulations and computational mathematics.

Abstract

Numerical investigations of partial differential equations with hysteresis have largely focused on simulations, leaving numerical error analysis unexplored and relying mainly on derivative-free nonlinear solvers. This work establishes rigorous finite element error estimates for the backward Euler fully discrete scheme applied to semilinear and quasilinear parabolic equations involving continuous hysteresis operators. To efficiently handle the inherent nonsmoothness of the resulting nonlinear algebraic systems, we develop a damped smoothing Newton solver under a general condition on the smoothing approximation, ensuring global convergence together with local Q-quadratic convergence. Numerical experiments confirm the theoretical convergence rates for semilinear problems, while showing higher-than-predicted orders for quasilinear ones. The robustness and efficiency of the proposed solver are further demonstrated in comparison with existing methods.

Parabolic hysteresis problems revisited: Finite element error analysis and convergent Newton-type solvers

TL;DR

This work advances the numerical analysis of parabolic PDEs with hysteresis by delivering rigorous finite element error estimates for backward Euler discretizations in both semilinear and quasilinear settings. It then introduces a robust Jacobian smoothing Newton framework, including an arc-based smoothing strategy, and proves global convergence with local quadratic rates. Comprehensive numerical experiments verify theoretical rates and demonstrate that the proposed arc-based solver outperforms classical derivative-free methods and other Newton-type approaches in speed and robustness. The results provide a practical, theoretically grounded pipeline for accurately and efficiently simulating hysteresis-influenced parabolic problems in 1–3D. The approach is applicable to a broad class of hysteresis models, including linear play and Preisach operators, with potential impact on engineering simulations and computational mathematics.

Abstract

Numerical investigations of partial differential equations with hysteresis have largely focused on simulations, leaving numerical error analysis unexplored and relying mainly on derivative-free nonlinear solvers. This work establishes rigorous finite element error estimates for the backward Euler fully discrete scheme applied to semilinear and quasilinear parabolic equations involving continuous hysteresis operators. To efficiently handle the inherent nonsmoothness of the resulting nonlinear algebraic systems, we develop a damped smoothing Newton solver under a general condition on the smoothing approximation, ensuring global convergence together with local Q-quadratic convergence. Numerical experiments confirm the theoretical convergence rates for semilinear problems, while showing higher-than-predicted orders for quasilinear ones. The robustness and efficiency of the proposed solver are further demonstrated in comparison with existing methods.

Paper Structure

This paper contains 18 sections, 14 theorems, 140 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Parabolic hysteresis problems
  3. Continuous hysteresis operators
  4. Parabolic equations with hysteresis
  5. Finite element error analysis
  6. Error analysis on semilinear equations with memory
  7. Error analysis on quasilinear equations with linear play hysteresis
  8. Convergent Jacobian smoothing Newton solvers
  9. Derivation of the abstract problem
  10. Global and local convergence
  11. An arc-based smoothing strategy
  12. Numerical experiments
  13. Convergence rate validation
  14. Solver performance
  15. Tangent extension
  16. ...and 3 more sections

Key Result

Proposition 2.10

\newlabelprop:preli_semi_pde_wp0 Let asp:preli_Op_Causalityasp:preli_Op_Continuityasp:preli_Op_Affinely_Boundednessasp:preli_Op_lipschitz hold. Given $u^0\in V$, $w^0 \in L^2\qty(\Omega)$ and $f\in L^2\qty(Q)$, there exists a unique weak solution $u$ to eq:preli_semi_pde_strong satisfying such that $u(\cdot, 0) = u^0$, $w:=\mathcal{W}\qty(u, w^{0}) \in L^2\left(\Omega ; C\qty[0, T]\right)$ and f

Figures (5)

  • Figure 1: Diagram of the linear play operator. (a) Input function $u(t)$. (b) Input-output $u$-$w$ curve exhibiting hysteresis.
  • Figure 1: Local smoothing approximation of the angle by an arc.
  • Figure 1: Initial meshes for convergence verification. (a) Two dimensional mesh ($N_{\mathrm{init}}=10$). (b) Three dimensional mesh ($N_{\mathrm{init}}=5$).
  • Figure 2: Benchmark for the Preisach model. (a) Excitation: $H(t) = 170\sin(4\pi t) + 170\sin(20\pi t + \pi)$. (b) Input-output $u$-$w$ curve.
  • Figure 3: Convergence comparison. (a) Residual at $t_{35}=0.109375$. (b) Outer iterations for the entire transient problem.

Theorems & Definitions (32)

  • Remark 2.3
  • Proposition 2.10
  • Proposition 2.11
  • Remark 3.3
  • Theorem 3.4
  • Proof 1
  • Lemma 3.5
  • Proof 2
  • Proposition 3.6: Monotonicity-type property
  • Proof 3
  • ...and 22 more